In Eq. (REF ), we have included a mass term to the 3-form field \(\Omega \) , which explicitly breaks the original gauge symmetry \(\Omega _{\mu \nu \rho } \rightarrow \Omega _{\mu \nu \rho } +3 \partial _{[\mu } b_{\nu \rho ]}\)  with \(b\)  a two-form gauge parameter field. Here we do not try to specify the possible origin of this mass term. Rather, we would like to emphasize its importance to the theory. Firstly, note that, without this mass term, the gauge symmetry involving \(b\)  is recovered in this Lagrangian of \(\Omega \) . Thus, the lightness of \(\Omega \)  is technically natural [1]}, which can be understood in connection with this underlying broken gauge symmetry. Secondly, further significance of this mass term can be seen by analyzing its particle content in \(\Omega \) . In the massless case, \(\Omega \)  does not possess any physical propagating degree of freedom (dof) in four spacetime dimensions [2]}. To see this, let us divide the components of \(\Omega \)  into two classes, \(\Omega _{0ij}\)  and \(\Omega _{ijk}\)  with the Latin indices denoting the spatial dimensions. On the one hand, due to the antisymmetric nature of the field strength, the Lagrangian in Eq. (REF ) cannot give rise to the kinetic term for \(\Omega _{0ij}\)  i.e., \(\partial _0 \Omega _{0ij}\)  cannot exist. On the other hand, \(\Omega _{ijk}\equiv \epsilon _{ijk} \phi \) , actually being a pseudo-scalar in nature, suffers from the gauge transformation as \(\Omega _{ijk} \rightarrow \Omega _{ijk} +3 \partial _{[i} b_{jk]}\) , so that it is unphysical either. However, in the model with a massive 3-form field, the gauge invariance of \(\Omega \)  is explicitly broken, so that \(\Omega _{ijk}\)  retains as true physical propagating dof in the theory. Following the tensor-projection operator technique developed in Ref. [3]}, [4]}, [2]}, [6]}, we can derive the propagator of the massive 3-form field \(\Omega \)  as follows
\(i\Delta ^\Omega _{\mu \nu \lambda ,\,\rho \sigma \kappa } (k) = \frac{-i{\cal P}_{\mu \nu \lambda \,,\rho \sigma \kappa }(k)}{k^2-m_\Omega ^2}\,, \) 
